1. Field of the Invention
Embodiments of the present invention relate to aperture-coded spectroscopy. More particularly, embodiments of the present invention relate to systems and methods for estimating the mean spectral density of a diffuse source in a single time step of parallel measurements using a static multimode multiplex spectrometer.
2. Background Information
A diffuse source is a source that inherently produces a highly spatially-multimodal optical field. In simplest terms, this is a spatially-extended source with an angularly-extended radiation pattern. The constant radiance theorem complicates the characterization of such sources. In short, entropic considerations require that the modal volume of a source cannot be reduced without a concomitant reduction in power. As a result, the brightness of diffuse sources cannot be increased.
This is particularly unfortunate in the case of spectroscopy, as traditional spectrometers utilize narrowband spatial filtering to disambiguate between spatial and spectral modes of the field. A dispersive element produces a wavelength-dependent shift of the image of an input slit. Since each spectral channel must correspond to a unique shift, the spectral width of a resolution element is directly proportional to the slit width.
This relationship provides a challenge to diffuse source spectroscopy. To achieve a reasonable spectral resolution, the input slit to the spectrometer must be narrow. However, because the source is diffuse, the radiation field cannot be focused through the slit. Instead, only a small fraction of the light can enter the instrument. If the source is weak as well as diffuse, then the instrument may be so photon starved that no spectral measurement is possible.
The throughput of an optical instrument, sometimes referred to as the etendue, can be approximated as the product of the area of the input aperture and the solid angle from which the instrument will accept light. The acceptance solid angle is determined by the internal optics of an instrument. For a given optical arrangement, the only way to increase the etendue of the system is to increase the size of the input aperture. However, such an approach reduces the resolution of the spectrometer as it increases the throughput.
Consequently, two primary challenges in diffuse source spectroscopy are maximizing spectrometer throughput without sacrificing spectral resolution and maximizing the signal-to-noise-ratio (SNR) of the estimated spectrum for a given system throughput and detector noise.
Both problems have been long-studied and a number of designs have been proposed to address one or both. A design that solves the first problem is said to have a Jacquinot (or large-area or throughput) advantage. A design that solves the second problem is said to have a Fellgett (or multiplex) advantage.
The earliest approach to solving these problems was through coded aperture spectroscopy, where the input slit is replaced with a more complicated pattern of openings. The first coded aperture spectrometer was created in the early 1950s. Advancements followed rapidly over the next several decades. As the mathematical treatments gained sophistication, the appeal of apertures based on Hadamard matrices became apparent, and the majority of coded aperture spectrometers became Hadamard transform (HT) spectrometers. Over most of their development, however, HT spectrometers had only single-channel detectors or limited arrays of discrete detectors. As a result, most designs contained at least two coding apertures, one at the input plane and one at the output plane. Further, the designs usually required motion of one mask with respect to the other. The majority of the resulting instruments exhibited only the Jacquinot advantage or the Fellgett advantage.
In a coded aperture spectrometer, a coded aperture or mask is used to convert intensity information into frequency or spectral information. The basic elements of a coded aperture imaging spectrometer are described in Mende and Claflin, U.S. Pat. No. 5,627,639 (the '639 patent), for example. Light from multiple locations on a target is incident on a mask. The mask contains rows and columns of both transmissive and opaque elements. The transmissive and opaque elements are located on the mask according to a transfer function used to convert intensity information of the incident light to spectral information. The transmissive elements transmit the incident light, and the opaque elements block the incident light. A grating is used to disperse the transmitted light from the transmissive elements in a linear spatial relationship, according to the wavelength of the transmitted light. The dispersed light is incident on a detector array. The detector array contains rows and columns of detector elements. The detector array elements are designed to receive a different range of wavelengths from each transmissive element of the mask and provide a signal indicative the intensity of the light received.
In the '639 patent, the mask is translated in one direction relative to the target over time. As the mask is translated, a data matrix is generated. The data matrix contains light intensity data from each row of the detector array as light incident from the same target elements passes through a corresponding row of the mask. The intensities recorded by the rows and columns of detector elements are collected over time and assembled in a data matrix for each set of target elements.
A frequency spectrum is obtained for each set of target elements by converting the data matrix according to the transfer function. In the '639 patent, a pattern matrix is predetermined from the mathematical representation of the mask elements. Transmissive elements of the mask are represented as a ‘1’ in the pattern mask, and opaque elements of the mask are represented as a ‘0’ in the pattern mask. A frequency matrix representing the frequency spectrum is obtained by multiplying the data matrix by the inverse pattern matrix and a factor that is a function of the number of transmissive elements and number of total mask elements.
Coded aperture spectroscopy was proposed in Golay, M. J. E. (1951), “Static multislit spectrometry and its application to the panoramic display of infrared spectra,” Journal of the Optical Society of America 41(7): 468-472. Two-dimensional coded apertures for spectroscopy were developed in the late 1950's and early 1960's as described, for example, in Girard, A. (1960), “Nouveaux dispositifs de spectroscopic a grande luminosite,” Optica Acta 7(1): 81-97.
For the first 40 years of coded aperture spectroscopy, coded aperture spectroscopy instruments were limited to single optical detector elements or small arrays of discrete detectors. Reliance on single detectors required mechanical, electro-optical, liquid crystal, or other forms of modulation to read spectral data. These early coded aperture instruments implemented spectral processing by using both an entrance and an exit coded aperture and a single detector element or a detector element pair.
High quality two-dimensional electronic detector arrays were in use by the 1990's, as described in the '639 patent, for example. Despite the availability of these high quality two-dimensional electronic detector arrays, coded aperture instruments combining entrance and exit coded apertures for two-dimensional codes are still in use today, as described, for example in Shlishevsky, V. B. (2002), “Methods of high-aperture grid spectroscopy,” Journal Of Optical Technology 69(5): 342-353.
The use of Hadamard codes in coded aperture spectroscopy was described in detail in Harwit, M. and N. J. A. Sloane (1979), Hadamard transform optics, New York, Academic Press, the subject matter of which is incorporated herein by reference. The '639 patent describes a two-dimensional Hadamard code design where the elements of each row of the mask are arranged in a Hadamard pattern, and each row of the mask has a different cyclic iteration of an m-sequence. A two-dimensional Hadamard code mask design with appropriately weighted row and column codes that form an orthogonal family is described F. A. Murzin, T. S. Murzina and V. B. Shlishevsky (1985), “New Grilles For Girard Spectrometers,” Applied Optics 24 (21): 3625-3630, for a spectrometer using both entrance and exit coded apertures and a single detector element or a detector element pair.
Aperture coding is not the only approach to solving these spectrometer design problems. Interferometric spectrometers, such as Fourier transform (FT) spectrometers, can also exhibit the Jacquinot and Fellgett advantages. The FT spectrometer, in fact exhibits both. However, the majority of FT spectrometers contain mechanical scanning elements.
In view of the foregoing, it can be appreciated that a substantial need exists for systems and methods that can advantageously provide for maximum spectrometer throughput without sacrificing spectral resolution and maximum SNR of the estimated spectrum for a given system throughput and detector noise.